3.809 \(\int \frac{x^{7/2} (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=286 \[ \frac{2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^3 \sqrt{x} (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-2*a^3*(A*b - a*B)*Sqrt[x]*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^2*(A*b - a*B)*x^(3/2)*(a + b
*x))/(3*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*a*(A*b - a*B)*x^(5/2)*(a + b*x))/(5*b^3*Sqrt[a^2 + 2*a*b*x + b
^2*x^2]) + (2*(A*b - a*B)*x^(7/2)*(a + b*x))/(7*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*B*x^(9/2)*(a + b*x))/(
9*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^(7/2)*(A*b - a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(b^(1
1/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.134835, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \[ \frac{2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^3 \sqrt{x} (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(-2*a^3*(A*b - a*B)*Sqrt[x]*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^2*(A*b - a*B)*x^(3/2)*(a + b
*x))/(3*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*a*(A*b - a*B)*x^(5/2)*(a + b*x))/(5*b^3*Sqrt[a^2 + 2*a*b*x + b
^2*x^2]) + (2*(A*b - a*B)*x^(7/2)*(a + b*x))/(7*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*B*x^(9/2)*(a + b*x))/(
9*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^(7/2)*(A*b - a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(b^(1
1/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{7/2} (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{x^{7/2} (A+B x)}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{7/2}}{a b+b^2 x} \, dx}{9 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{5/2}}{a b+b^2 x} \, dx}{9 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 a^2 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{a b+b^2 x} \, dx}{9 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a^3 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{9 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x} (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 a^4 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{9 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x} (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (4 a^4 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x^2} \, dx,x,\sqrt{x}\right )}{9 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x} (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.083478, size = 139, normalized size = 0.49 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} \left (21 a^2 b^2 x (5 A+3 B x)-105 a^3 b (3 A+B x)+315 a^4 B-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )-315 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{315 b^{11/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(A + B*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*(a + b*x)*(Sqrt[b]*Sqrt[x]*(315*a^4*B - 105*a^3*b*(3*A + B*x) + 21*a^2*b^2*x*(5*A + 3*B*x) - 9*a*b^3*x^2*(7
*A + 5*B*x) + 5*b^4*x^3*(9*A + 7*B*x)) - 315*a^(7/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(315*b
^(11/2)*Sqrt[(a + b*x)^2])

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 197, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{315\,{b}^{5}} \left ( 35\,B\sqrt{ab}{x}^{9/2}{b}^{4}+45\,A\sqrt{ab}{x}^{7/2}{b}^{4}-45\,B\sqrt{ab}{x}^{7/2}a{b}^{3}-63\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+63\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+105\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}-105\,B\sqrt{ab}{x}^{3/2}{a}^{3}b-315\,A\sqrt{ab}\sqrt{x}{a}^{3}b+315\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{4}b+315\,B\sqrt{ab}\sqrt{x}{a}^{4}-315\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)/((b*x+a)^2)^(1/2),x)

[Out]

2/315*(b*x+a)*(35*B*(a*b)^(1/2)*x^(9/2)*b^4+45*A*(a*b)^(1/2)*x^(7/2)*b^4-45*B*(a*b)^(1/2)*x^(7/2)*a*b^3-63*A*(
a*b)^(1/2)*x^(5/2)*a*b^3+63*B*(a*b)^(1/2)*x^(5/2)*a^2*b^2+105*A*(a*b)^(1/2)*x^(3/2)*a^2*b^2-105*B*(a*b)^(1/2)*
x^(3/2)*a^3*b-315*A*(a*b)^(1/2)*x^(1/2)*a^3*b+315*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^4*b+315*B*(a*b)^(1/2)*x^(1
/2)*a^4-315*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5)/((b*x+a)^2)^(1/2)/b^5/(a*b)^(1/2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.36375, size = 617, normalized size = 2.16 \begin{align*} \left [-\frac{315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{315 \, b^{5}}, -\frac{2 \,{\left (315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}\right )}}{315 \, b^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/315*(315*(B*a^4 - A*a^3*b)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(35*B*b^4*x^4
+ 315*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x^3 + 63*(B*a^2*b^2 - A*a*b^3)*x^2 - 105*(B*a^3*b - A*a^2*b^2
)*x)*sqrt(x))/b^5, -2/315*(315*(B*a^4 - A*a^3*b)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) - (35*B*b^4*x^4 + 315
*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x^3 + 63*(B*a^2*b^2 - A*a*b^3)*x^2 - 105*(B*a^3*b - A*a^2*b^2)*x)*
sqrt(x))/b^5]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x+A)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.20385, size = 277, normalized size = 0.97 \begin{align*} -\frac{2 \,{\left (B a^{5} \mathrm{sgn}\left (b x + a\right ) - A a^{4} b \mathrm{sgn}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{2 \,{\left (35 \, B b^{8} x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) - 45 \, B a b^{7} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + 45 \, A b^{8} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + 63 \, B a^{2} b^{6} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) - 63 \, A a b^{7} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) - 105 \, B a^{3} b^{5} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 105 \, A a^{2} b^{6} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 315 \, B a^{4} b^{4} \sqrt{x} \mathrm{sgn}\left (b x + a\right ) - 315 \, A a^{3} b^{5} \sqrt{x} \mathrm{sgn}\left (b x + a\right )\right )}}{315 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

-2*(B*a^5*sgn(b*x + a) - A*a^4*b*sgn(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) + 2/315*(35*B*b^8*x
^(9/2)*sgn(b*x + a) - 45*B*a*b^7*x^(7/2)*sgn(b*x + a) + 45*A*b^8*x^(7/2)*sgn(b*x + a) + 63*B*a^2*b^6*x^(5/2)*s
gn(b*x + a) - 63*A*a*b^7*x^(5/2)*sgn(b*x + a) - 105*B*a^3*b^5*x^(3/2)*sgn(b*x + a) + 105*A*a^2*b^6*x^(3/2)*sgn
(b*x + a) + 315*B*a^4*b^4*sqrt(x)*sgn(b*x + a) - 315*A*a^3*b^5*sqrt(x)*sgn(b*x + a))/b^9